Neal (1996) proved that infinitely wide shallow Bayesian neural networks (BNN) converge to Gaussian processes (GP), when the network weights have bounded prior variance. Cho & Saul (2009) provided a useful recursive formula for deep kernel processes for relating the covariance kernel of each layer to the layer immediately below. Moreover, they worked out the form of the layer-wise covariance kernel in an explicit manner for several common activation functions. Recent works, including Aitchison et al. (2021), have highlighted that the covariance kernels obtained in this manner are deterministic and hence, precludes any possibility of representation learning, which amounts to learning a non-degenerate posterior of a random kernel given the data. To address this, they propose adding artificial noise to the kernel to retain stochasticity, and develop deep kernel inverse Wishart processes. Nonetheless, this artificial noise injection could be critiqued in that it would not naturally emerge in a classic BNN architecture under an infinite-width limit. To address this, we show that a Bayesian deep neural network, where each layer width approaches infinity, and all network weights are elliptically distributed with infinite variance, converges to a process with $\alpha$-stable marginals in each layer that has a conditionally Gaussian representation. These conditional random covariance kernels could be recursively linked in the manner of Cho & Saul (2009), even though marginally the process exhibits stable behavior, and hence covariances are not even necessarily defined. We also provide useful generalizations of the recent results of Lor\'ia & Bhadra (2024) on shallow networks to multi-layer networks, and remedy the computational burden of their approach. The computational and statistical benefits over competing approaches stand out in simulations and in demonstrations on benchmark data sets.

本研究解决了深度核过程中的确定性协方差核导致的表示学习问题，提出了一种新颖的方法，通过无限方差的贝叶斯深度神经网络来获得条件高斯表示的随机协方差核。关键发现是该方法在计算和统计上相较于现有方法具有显著优势，尤其在模拟和基准数据集展示效果显著。

无限方差先验权重下的深核后验学习